Let us fix any \(\ast \) -subalgebra \(\overline{\mathcal {O}}(\mathbb {R}) \subseteq \mathrm {Fun}(\mathbb {R},\mathbb {C})\) 
of the \(\ast \) -algebra of complex-valued functions on the real line that contains the
monomials \(y^n\in \overline{\mathcal {O}}(\mathbb {R})\) , for \(n\ge 0\) ,
and also the exponential functions \({\rm e}^{{\rm i}c y}\in \overline{\mathcal {O}}(\mathbb {R})\) , for \(c\in \mathbb {R}\) .
For example, we could choose the algebra of analytic functions \(C^\omega (\mathbb {R},\mathbb {C})\) ,
the algebra of smooth functions \(C^\infty (\mathbb {R},\mathbb {C})\)  or the algebra of continuous functions \(C^0(\mathbb {R},\mathbb {C})\) . Note that the polynomial algebra \(\mathbb {C}[y]\)  used in [1]}
does not satisfy our criteria because it does not contain the exponential functions.
Consider the two evaluation maps \(\mathrm {ev}_{p}\colon \overline{\mathcal {O}}(\mathbb {R})\rightarrow \mathbb {C}\) ,
\(f\mapsto f(p)\) , for \(p=0,1\) , and note that they are algebra homomorphisms.
The following definition generalizes [1]}
to the new concept of \(\overline{\mathcal {O}}(\mathbb {R})\) -homotopy equivalence.
